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\begin{document}
%% DAY 1
\begin{center}
${\bf 42\nd}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Washington, D.C., United States of America} \\ [.05in]
{\bf Day I \hspace{.25in} 9 a.m. - 1:30 p.m.}\\[.05in]
{\bf July 8, 2001}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\item %% IMO1
Let $ABC$ be an acute-angled triangle with $O$ as its circumcenter. Let
$P$ on line $BC$ be the foot of the altitude from $A$. Assume that $\ang
BCA \ge \ang ABC + 30\dg$. Prove that $\ang CAB + \ang COP < 90\dg$.
\item %% IMO2
Prove that
\[
\frac{a}{\sqrt{a^2+8bc}} + \frac{b}{\sqrt{b^2+8ca}} +
\frac{c}{\sqrt{c^2+8ab}} \ge 1
\]
for all positive real numbers $a, b$, and $c$.
\item %% IMO3
Twenty-one girls and twenty-one boys took part in a mathematical
competition. It turned out that
\be
\ii[(a)]
each contestant solved at most six problems, and
\ii[(b)]
for each pair of a girl and a boy, there was at least one problem that
was solved by both the girl and the boy.
\ee
Prove that there is a problem that was solved by at least three girls and
at least three boys.
\end{enumerate}
\pagebreak %% DAY 2
\begin{center}
${\bf 42\nd}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Washington, D.C., United States of America} \\ [.05in]
{\bf Day II \hspace{.25in} 9 a.m. - 1:30 p.m.}\\[.05in]
{\bf July 9, 2001}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\setcounter{enumi}{3}
\item %% IMO4
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$
be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of
$\{\,1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that
there exist permutations $b$ and $c$, $b \ne c$, such that $n!$ divides
$S(b)-S(c)$.
\item %% IMO5
In a triangle $ABC$, let segment $AP$ bisect $\ang BAC$, with $P$ on side
$BC$, and let segment $BQ$ bisect $\ang ABC$, with $Q$ on side $CA$. It
is known that $\ang BAC = 60\dg$ and that $AB + BP = AQ + QB$. What are
the possible angles of triangle $ABC$?
\item %% IMO6
Let $a > b > c > d$ be positive integers and suppose
\[
ac + bd = (b+d+a-c)(b+d-a+c).
\]
Prove that $ab + cd$ is not prime.
\end{enumerate}
\end{document}